Spectral dimension of a quantum universe

TL;DR

This paper calculates the spectral dimension of quantum spacetime, revealing a scale-dependent behavior where the dimension transitions from 4 to 2 at minimal length scales, indicating potential renormalizability of gravity.

Contribution

It introduces a transparent method to compute the spectral dimension considering quantum fluctuations and minimal length effects, extending analysis to curved backgrounds.

Findings

Spectral dimension approaches 2 at minimal length scale.

At large scales, spacetime behaves like a smooth 4D manifold.

Spectral dimension tends to zero at extremely small diffusion times.

Abstract

In this paper, we calculate in a transparent way the spectral dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a spectral dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the spectral dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the spectral dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.